Reverberation chambers are now common test facilities used in electromagnetic compatibility. Objects loaded in the chambers, such as antennas and reflectors, can affect the measurement because they absorb and scatter the diffuse field. Thus, it is important to know the scattering and absorbing cross sections of these objects. In this paper, we propose a new technique to measure the scattering cross sections averaged over all angles of incidence and both polarizations in a reverberation chamber. It is based on stacking time-dependent fields generated by a repetitive pulsed source while the objects are moving. The fields recorded by wire antennas are averaged over the object positions. The square of the averaged field is shown to decrease exponentially with the time spent by the wave in the chamber. The cross section is deduced from the exponential factor. The technique has been experimentally validated in a 1 m3 reverberation chamber at 2.45 GHz with metallic spheres of several radii. The experimental cross sections are found to be in agreement with the theoretical ones.